247 dynamic cutting process modelling and its impact on

Dynamic cutting process modelling and its impacton the generation of surface topography andtexture in nano/micro cuttingL Zhou1,2* and K Cheng2

1School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, Heilongjiang, People’s Republic of

China2School of Engineering and Design, Brunel University, Uxbridge, UK

The manuscript was received on 1 August 2008 and was accepted after revision for publication on 26 November 2008.

DOI: 10.1243/09544054JEM1316

Abstract: In the nano/micro cutting process, the surface quality is heavily dependent on all thedynamic factors, including those from the material, tooling, process parameters, servo accuracy,mechanical structural stiffness, and non-linear factors as well. The machined surface is generatedbased on the tool profile and the real tool path combining with the various external and internaldisturbances. To bridge the gap between the cutting process and the surface topography/texturegeneration, an integrated simulation-based approach is presented involving the dynamic cuttingprocess, control/drive system, and the surface generation. The simulations take account of all theintricate aspects of the cutting process resulting in the surface topography and texture formation,such as material heterogeneity, regenerative chatter, built-up edge (BUE), tool wear, spindle run-out, environmental vibration, tool interference, etc. Both the frequency ratio method andsampling theorem are used to interpret the surface topography and texture formation. The effectsof non-linear factors on the surface generation are simulated and analysed through the powerspectral density (PSD) and significance on surface texture. The relationships among cutting force,tool path, and surface profile are discussed in detail. Furthermore, the proposed systematicmodelling approach is verified by cutting trials, which provide the coincident results of thesurface topography and areal power spectral density (APSD).

Keywords: dynamic cutting force, surface topography, surface texture, tool interference,APSD, significance analysis

1 INTRODUCTION

Metal cutting at the nano/micro scale is a complexprocess that comprises the workpiece material,tooling geometry, cutting process parameters, servodrive accuracy, static and dynamic deformations ofthe machine structure, etc. The machined surface isgenerated based on the tool profile and real tool pathconsidered with various disturbances. It is of greatbenefit to develop a systematic model and analysethe effects from the dynamic cutting process. Manylinear/non-linear factors in the whole system will

contribute to the surface topography and textureformation, such as material heterogeneity [1, 2],chatter [3], built-up edge (BUE) [4], tool wear [5],spindle runout [6], environmental vibration, etc. Thedynamic cutting force reflects the collective effects ofall the factors above, and leads to the tool–workpiecerelative displacement in the stiffness loop. Therefore,the investigation on the dynamic cutting force mod-elling and its impact is essential and much needed,particularly in a scientific and analytical manner.

Taking the three-axis turning machine as a casestudy in point, as shown in Fig. 1, the stiffness loop inthe face cutting process involves:

(a) spindle axial runout,(b) workpiece clamp error,(c) workpiece material property,

*Corresponding author: School of Mechatronics Engineering,

Harbin Institute of Technology, New Technology Building,

No 92, West Da Zhi Street, Harbin, Heilongjiang 150001,

People’s Republic of China. email: [email protected]

JEM1316 � IMechE 2009 Proc. IMechE Vol. 223 Part B: J. Engineering Manufacture

247

(d) cutting tool geometry,(e) tool holder stiffness,(f) Z axis servo stiffness in the axial direction,(g) X axis slideway stiffness in the side direction,(h) mounting stiffness.

In the nano/micro cutting process, the surfaceperformance of undertaking the high-precisionmachining is all about how the tool–workpiecerelative position rightly maintains its desired objec-tive. In the machine stiffness loop, all the elementsare guaranteed by physical construction like bear-ings, mounting screws, and other mechanical struc-tures except the linear motor in the axial direction.Since employing the direct driven linear motor, thestiffness in Z axial motion is directly provided by the

electrical servo motor. The tuning of the Z axis linearmotor, which is in the error-sensitive direction, istherefore one of the most critical ingredients ofdesign and uses the ultra-precision machine tool.

The topography and texture of the machined sur-face are generated by the dynamic feed responseunder the multifarious cutting factors. Several litera-tures focus on the surface topography generationprocess in relation to getting the connection of thetool path and the machined surface. Cheung and Lee[7–9] and Kim et al. [10] used the digitizing oscillo-scope to record the relative displacement between thecutting tool and the workpiece surface. After beinganalysed by fast Fourier transform (FFT), the mainfrequency and amplitude of the vibration reconstructsthe tool deviation from the ideal position. The tool

Fig. 1 The machine control loop and stiffness loop in nano/micro cutting

Proc. IMechE Vol. 223 Part B: J. Engineering Manufacture JEM1316 � IMechE 2009

248 L Zhou and K Cheng

profile superimposes on to the tool path to generatethe surface topography. Lee and Cheung and collea-gues [11, 12] developed a dynamic cutting system topredict the surface topography in face turning.However, the cutting force model only contains thelinear factors, and the whole response system wassimplified into a second-order system. In fact, theadvanced control algorithms are always applied toultra-precision machine tools so as to achieve a higherservo performance. Using a second-order systemmay marginally succeed in traditional machine tools,but it loses the modelling accuracy in ultra-precisionmachines. The servo control system, material hardgrain effect, chatter, tool wear, spindle runout, andenvironmental vibration were also not taken intoaccount in these papers. Luo et al. [13, 14] introducedsome non-linear factors into the cutting process andcontinued to use a second-order response system.Furthermore, the effect of non-linear factors on thesurface generation had not been involved.

For the generation algorithm of the machined sur-face, themost important procedure is to get the correctremaining profiles. The traditional approaches [8–13]compute the intersection of two adjacent tool profilesin order to get each tool profile boundary. However,single-point diamond turning, which is under a finerfeed rate, a larger tool nose radius, a higher spindlerotational speed, and a greater unit cutting force, willbe more susceptible to the effect of the tool inter-ference [7, 15]. The computation of remaining profileson the machined surface should be based on not onlythe neighbouring tool profiles but also all the profilesthat might intersect and overcut the previous toolmarks because of the tool interference effect.

In this paper, the following novel work is pre-sented, including:

1. To develop the control/drive system model intothe systematic prediction model of the surfaces inface turning.

2. To include the non-linear factors into the dynamiccutting model to simulate the real dynamiccutting process and study the significance of thefactors on surface quality.

3. To produce a surface generation algorithm inconsideration of the tool interference effect.

4. To elucidate the formation of the flutes/ringson the machined surface and find out the errororigins of the dynamic cutting factors throughthe power spectral density (PSD) analysis.

2 SYSTEMATIC MODELLING OF THE DYNAMICNANO/MICRO CUTTING

As illustrated in Fig. 2, the generation of the work-piece surface in nano/micro cutting is a very complexmaterial removal process, including:

(a) servo control dynamics, which is dependent oncontrol algorithms, servo amplifiers, actuators,and inspection sensors;

(b) machining process dynamics, which is heavilyaffected by the machine tool performance, tool-ing geometry, workpiece material properties,operation factors, servo motion, and workingenvironment;

(c) surface generation and surface analysis on itstexture, integrity, and functionality.

To obtain a better understanding of this complexprocess, a systematic model is developed in theMATLAB Simulink module, as shown in Fig. 3. Themodelling approach bridges the gap between thedeterminative factors and the surface generation.Based on a thorough theoretical analysis of servomotions and cutting mechanics and dynamics, theintegrated model produces a scientific methodologyto simulate the precision surface generation innano/micro turning processes.

Fig. 2 An illustration of the machining process integration system


Dynamic cutting process modelling 249

Fig.3

Integratedmodellingofthedynamic

cuttingprocess



2.1 Modelling of servo control dynamics

Most of the simulations on cutting dynamics are sim-plified by modelling the control and drive system as asecond-order system. However, the details in the servocontrol system cannot be neglected. In this paper,the ultra-precision turning machine presented em-ploys the UMAC motion controller, which provides aproportional-integral-derivative (PID) position/velo-city servo control scheme with velocity and accelera-tion feedforward loops as shown by [16]

DACout ¼2�19Kp Ksp FE þ KvffCV þ KaffCA

128þ KiIE

223

� ��

� KdKsvAV

128

�ð1Þ

A PID loop with feedforward compensation is used tocontrol all the machine axes. The feedforward assistsin eliminating most of the time delay between com-manded and actual positions. In other words, itreduces the phase lag in the system. An augmentablefirst-order lowpass filter helps to attenuate thequantization noise, particularly when higher band-widths are required.

The block diagram in Fig. 4 strictly follows equa-tion (1) to simulate the real controller activities. Thediscrete model uses Z-transforms whose samplingperiod is the real servo interrupt cycle. The positionquantification based on the encoder resolution is4.88 nm. A linear compensation function counteractsthe friction from the physical plant according to thevelocity direction.

The output from the UMAC controller is led tothe servo amplifier to continue the current loop, asshown in Fig. 5. The direct-drive linear motor model isbuilt on the resistance and inductance of the motorwindings, force constant, and the back electromotiveforce (EMF) constant from the specifications. In thiscase, the aerostatic bearing slideway is applied toreduce the friction and unexpected load in the motiondirection. Therefore, the plant of the slide carriage issimplified to represent a pure mass module.

From the servo control dynamics model, the actualmotion of each slideway is calculated for the next stage

to generate the real tool path. In fact, there are someadditional disturbances in the drive system, such as thecutting force disturbance superimposed on the slide-ways, which is generated by the dynamic cutting process.

2.2 Dynamic cutting process

The quality of the machined surface is mainlydetermined by the relative motions between the cut-ting tool edge and the workpiece. In practice, the toolpath deviates from the ideal tool path as a result ofkinematic or dynamic factors from the dynamic cut-ting process, machine tool motion errors, and envir-onmental disturbances. For instance, the deformationof the aerostatic slideway under the action of dynamiccutting forces will deflect the cutting tool nose point;environmental vibrations will move the motion of thecutting tool away from the designed tool path.

Since the surface topography model basically fol-lows the reflection of the cutting tool edge profile, thecalculation of the real tool path becomes the criticalpart in the modelling approach. Table 1 lists the linearand non-linear factors from the structural deformationandmotion errors of themachine tool and cutting tool,which will contribute to the deviation of the tool path.

All the cutting process dynamics related to func-tions above are modelled as illustrated in Fig. 6. Inthe same way, the motion errors of the machine toolpresented in Fig. 3 are superimposed on the output ofthe slideways to generate the real tool path.

As demonstration workpiece material is aluminiumalloy, the dynamic cutting force model will thus followthe elastic–plastic mechanics model. According to theelastic–plastic deformation principle, the forces actingon the tool rake face can be acquired by the coordinatetransformation of the shear plane force based on theshear plane cutting model. The force acting on the toolcutting edge and flank face can be deduced based onthe empirical formula of the contact stress and elasticrecovery. Accumulating the forces’ action on the threezones, there will be the dynamic cutting forces in thethree directions, following the Cartesian coordinatesystem in Fig. 1, expressed as [19, 20]

FxðtÞ ¼ ðKfc � KrcÞ sin urðdc þ DctÞ ðh þ DcwÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR20 � ðR0 � dc � DctÞ2

q� �þ KreR0ur sin ur

FyðtÞ ¼ Ktcðdc þ DctÞ ðh þ DcwÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR20 � ðR0 � dc � DctÞ2

q� �

þ KteR0ur þ Kff1 sinb

2� m cos

b

2

� �þ Kff2ðsinb � m cosbÞ

FzðtÞ ¼ ðKfc sin ur þ 1 � cos urÞðdc þ DctÞ ðh þ DcwÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR20 � ðR0 � dc � DctÞ2

q� �

þ ðKfe sin ur þ 1 � cos urÞR0ur þ Kff1 cosb

2þ m sin

b

2

� �þ Kff2ðcosb þ m sinbÞ ð10Þ



Fig.4

Servocontrollermodel



Fig.5

Drivesystem

model



As described, the real tool path is created by all theabove factors, including servo motions influenced bydynamic cutting disturbance, machine tool defor-mations, and environmental vibrations. Therefore, itis expressed as

xtp ¼ xsm þ eevx þ esr � Fx=Kzx

ztp ¼ zsm þ eevz þ esa � Fz=Kxz

ð11Þ

3 SIMULATION OF NANO/MICRO MACHINEDSURFACES

3.1 Prediction algorithms for a machined surface

The ultimate stage of the machining system is torender the surface topography. The cutting tool willfollow the real tool path to reproduce the tool profileon the machined surface in the form of feed marks, asillustrated in Fig. 7.

The traditional surface generation algorithms[8–13] calculate the intersection of two adjacent toolprofiles to get the boundary of each feed mark, as inFig. 8(a). However, in the ultra-precision machining,the feedrate is so small that the next several cuts willclean up the previous tool marks when there arevibrations inmotion. As shown in Fig. 8(b), the (iþ 1)thand (iþ 2)th tool profiles, which are overcut by the(iþ 3)th, cannot affect the workpiece surface gen-eration. The machined surface is contoured by theith, (iþ 3)th, (iþ 4)th, etc. tool profiles as a result oftool interference from vibrations. Generally speaking,the vibration of the tool path is not the entire reason

for forming the surface topography. The tool inter-ference appears normally in the cutting process witha small feedrate, which always helps to filter out thetool–workpiece vibration and form a more complexsurface topography.

To simulate the three-dimensional surface topo-graphy, the surface is sampled into a finite numberof equally spaced radial sections. The number ofsections, Ns, can be expressed as

Ns ¼ 2p=Du ð12Þand the number of spindle rotation rounds duringmachining, Nr, can be calculated as

Nr ¼ Rw=f ð13ÞTherefore, the total simulation points are

Np ¼ NsNr þ 1 ð14ÞThe dynamic cutting process model outputs xtp andztp to form the real tool path as the ultimate result. Inthe face-turning situation with cylindrical coordi-nates, xtp is actually the radius of the tool nose pointand ztp the amplitude correspondingly. To convertthe three-dimensional tool path into a rectangularcoordinate system, there are

xc ¼ xtp cosðpDuÞyc ¼ xtp sinðpDuÞ ðp ¼ 1; 2; . . . ;NpÞzc ¼ ztp

ð15Þ

Table 1 Linear and non-linear factors in the cutting process

Source Influence factor Mathematical function

BUE Rake angle Da¼aA Saw (vat) (2)

Hard grain Shear stress andhardness of workpiece

Dt¼At Pul(t) (3)DH¼AHPul(t)

Shear angle oscillation Shear angle Du¼Au sin(2pfut) (4)

Flank wear Tool wear lw ¼ AHt

FrVf Vs þ B exp �E

RTf

� �½17; 18� (5)

Regenerative vibration Feed and depth of cutchatter

Dct¼ z(t)� z(t�T) (6)

Dcw¼ x(t)� x(t�T)

Spindle runout Displacement betweenthe cutting tool and theworkpiece

esr¼Asr sin(vtþfr) (7)esa¼Asa sin(vtþfa)

Slideway stiffness ezx¼Fx/Kzx (8)exz¼Fz/Kxz

Environmental vibration evx¼Aevx sin(2pfevt) (9)evz¼Aevz sin(2pfevt)



As shown in Fig. 8, tool profiles in each radial sectionmay interfere with the earlier ones. For this reason, itis better to calculate all the tool profiles according tothe tool nose points and maintain the lowest curves.At the jth radial section, the tool nose point in thepolar coordination form can be described as

uaði; jÞ ¼ ðj � 1ÞDuraði; jÞ ¼ xtp½ði � 1ÞNs þ j�haði; jÞ ¼ ztp½ði � 1ÞNs þ j�

i ¼ 1; 2; . . . ;Nr; j ¼ 1; 2; . . . ;Ns

ð16Þ

Fig. 6 Cutting process dynamics model



To superpose the tool profiles on to the tool nosepoints, the sampling points at the jth radial sectionshould be refined by Df, which is smaller than f toshape the tool edge. Setting the tool nose radius as R0

utðk; i; jÞ ¼ ðj � 1ÞDurtðk; i; jÞ ¼ raði; jÞ þ kDf

htðk; i; jÞ ¼ haði; jÞ þ R0 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR20 � ðkDf Þ2

q� �

� Lt

2Df6 k<

Lt

2Df

ð17Þ

and the tool projected length in the X direction as Lt,the height of the surface point at the (i, j ) point isgeometrically determined as

Comparing all the tool edge profiles with the depthof cut dc, it is easy to get the residual contour whilethe relevant radius is calculated by the linear inter-polation method, i.e. there is

uðl; jÞ ¼ ðj � 1ÞDurðl; jÞ ¼ raði; jÞ þ ðl � if =Df Þ½raði þ 1; jÞ � raði; jÞ�hðl; jÞ ¼ min½htðk; i; jÞ;dc�

l ¼ 1; 2; . . . ;Rw=Df ð18Þ

Fig. 7 Simulation of feed marks on the machined surface

Fig. 8 Simulations on the two-dimensional machined surface profiles



Transforming the polar coordinates to Cartesianones, the three-dimensional surface topography isexpressed as

x ¼ r cosðuÞy ¼ r sinðuÞz ¼ h ð19Þ

All the three-dimensional surface topography pre-diction procedures are described as a block flowdiagram in Fig. 9. The principal advantage of thisapproach is in considering multiple tool interference,which is more consistent than the real practice.

3.2 Simulation on the surface generation

The relative displacement between the tool and theworkpiece in the Z direction is mainly decided bythe workpiece material, machine tool performance,tooling geometry, and cutting parameters, which arelisted in Table 2. The simulated surface, cut from theedge to the centre of the workpiece, is a 1.4mm· 1.4mm square located at the spindle central line, asshown in Fig. 10.

The impressive features on the surfaces are therings and flutes, which are much bigger than the toolfeed mark size. To analyse the waviness structureregulation, it is essential to work out the intrinsicconnection between the vibrations and the machinedsurface topography.

Since the tool paths in one section are the discretepoints by feed spacing per spindle revolution, thesurface generation acts as the process of the cuttingtool edge sampling the vibratory tool path in the feeddirection. The machined surface topography undervibration is directly determined by the relationbetween the vibration frequency fv and the spindlerotational frequency fs, i.e. their ratio fr is

fr ¼ fvfs

¼ fvn=60

¼ a þ b ð20Þ

In this case namely, fr is the frequency that the tooltip traverses the vibration within one spindle rotationperiod. The number of flutes is identical to the inte-ger a, which is the number of times the vibration iscompletely undergone per spindle revolution. Theabsolute fractional value |b|, which cannot cover thefull vibration cycle, will make the phase offset in eachspindle rotation. The orientation of the flutes isdecided by the sign of b. A positive sign meanscounterclockwise and a negative sign is clockwise inthis case. In other words, the radial section representsthe fractional part b of the frequency ratio, while theinteger a will be reflected in the circumferentialdirection.

4 SIMULATION RESULT ANALYSIS

As described in section 3, the machined surfacegeneration can be geometrically thought of as a resultof transferring the tool profile on to the workpiece. Toexplore the surface topography and texture char-acteristic further, it is essential to find out the sourcecausing the tool–workpiece relative deflection.

4.1 Power spectral analysis

The fast Fourier transform (FFT) function is exten-sively used to study the frequency components of asignal buried in a noisy time domain signal. Thepower spectral density (PSD) is intended for discretespectra computation with the FFT algorithm. Theintegral of the PSD over a given frequency bandcomputes the average power in the signal in thatfrequency band. The peak of the PSD at a given fre-quency reflects the power per unit frequency. Inother words, the signal’s frequency contents andtheir distribution at the frequency can be directlyperceived through the PSD analysis.

From the PSD of Fz in Fig. 11, it is found that thecutting force in the Z direction is mainly focused on25Hz (BUE), 32Hz (hard grain), 45Hz (shear angleoscillation), and their high-order harmonic compo-nents. From the amplitudes of the PSD, the mostcontributions in power are clearly from workpiecematerial properties in contrast with the others fromthe cutting process.

It is shown in Fig. 12 that all the frequency com-ponents of the cutting force clearly remain on thetool–workpiece relative displacement in the dynamiccutting process. The differences in the PSD from Fzare an additional 3.47Hz (environmental vibration)and 16.67Hz (spindle runout). The cutting forcedisturbance will be rejected by the servo motor andbearings on the slideway. From Fig. 12, even with thesmall force turbulence, the direct drive system willlose much accuracy. To improve the resistance of thedisturbance, it is better to increase the damping ofthe servo control system and the physical stiffnessand damping of slideways.

Figure 13 presents the tool path, tool profiles,and the final resultant surface topography in onesection. When converting the tool path into the sur-face topography, the temporal frequency fv (Hz)should also be converted to the spatial frequency fvs(mm�1) as

fvs ¼ fvfsf

ð21Þ

Accordingly, the key frequencies in the tool path of3.5, 16.7, 25, 32, 45, 64, and 96Hz will be equal to 21,100, 150, 192, 270, 384, and 576mm�1. In the light of



equations (20) and (21), |b| in the ascending sort is0.08, 0.16, 0.21, 0.24, 0.3, 0.5, and 1, whose corre-sponding spatial frequencies are 8, 16, 21, 24, 30, 50,and 100mm�1. The values closely dovetail with thePSD frequencies, as shown in Fig. 13.

Viewing from the frequency domain, the surfacetopography in one section is sampled by the feedingdistance per spindle revolution. In fact, the cutting inthe radial section is the process to sample the toolpath by the spindle rotational frequency fs. Since thetool path contains many frequencies exceeding fs/2,the process will not conform to the Nyquist conditionof the sampling theorem. Aliasing will occur and the

Fig. 9 A flowchart of the simulation model to predict the three-dimensional machined surface topo-graphy

Table 2 Simulation conditions for face turning

Factors Frequency Amplitude Function

Workpiece material(Aluminium 6061)

BUE 25Hz 2� Sawtooth wave

Hard grain – shear stress 32Hz 30MPa Square wave– hardness 32Hz 0.1GPa Square wave

Shear angle oscillation 45Hz 2� Sine wave

Machine toolperformance

Environmental vibration 3.5Hz 10nm Sine waveSpindle axial runout 16.67Hz 20nm Sine waveSlideway side stiffness — 40N/mm —

Tooling geometry Tool nose radius — 0.508mm —Cutting edge radius — 50nm —Top rake angle — � 2.5� —

Cutting parameters Spindle rotational speed — 1000 r/min —Feedrate — 0.01mm/rev —Depth of cut — 0.01mm —

Fig. 10 Simulation on the face turning surface



subsampling effect aliases the high-frequency signalsinto low-frequency ones. On this account, the newfrequency spectrum fp is proposed within one sam-pling bandwidth (� fs/2< fp 6 fs/2)

fp ¼ mfs � fv;fvfs

� 1

2<m6 fv

fsþ 1

2ð22Þ

The sampling method leads to an identical result asto the frequency ratio explained above.

At the same time, the higher frequency vibrationsare attenuated rapidly, which means the wavelengthsof the waviness produced at the radial sections are allgreater than f/2. Therefore, the waviness in the radialdirection can be controlled by the spindle speed andfeed rate.

4.2 Areal power spectral analysis

Although PSD can explain the formation of thesurface profile in two dimensions in one radial sec-tion, it is incapable of evaluating the whole surfacetopography. However, the trend in modern advan-ced manufacturing is to obtain the functionality ofa component by the control of three-dimensionaltopography and texturing. The areal method in three

dimensions is much more powerful than the profilemethod in two dimensions, in that it can be usedto characterize functional aspects of a surfacetopography. Furthermore, it shows the intricateembedded information to how the surface wasmanufactured.

The areal power spectral density (APSD) is a three-dimensional analysis that renders the information onthe overall surface topography. Figure 14 shows theAPSD of the face-turning surface which has stronganisotropy in the right-angled axes but obviousisotropy in the polar coordinates. The peaks are dis-tributed on specific frequency rings, which are iden-tical to those on the previous PSD components. Thesections on the same radius point out the numberof flutes on the machined surface. In other words,the number of sections in APSD is represented as m(or a), as explained before, and the radius states thespatial frequency fp (or b). Taking the 50mm�1 ringas an example, the ring is separated into four centre-symmetric sections. Owing to the dual symmetry ofthe APSD function, the four peaks give m¼ 2.Therefore, the original vibration should be 150mm�1,which comes from a 25Hz BUE triangular wave.

Fig. 11 Cutting force in the Z direction and its PSD



Fig. 13 Surface profile and its PSD

Fig. 12 Tool–workpiece relative displacement in the Z direction and its PSD



In Fig. 10, there are two apparent counterclockwiseflutes at the centre region of the surface. The wave-length of the flute is roughly 0.1–0.2mm/cycle; inother words, the spatial frequency is 5–10mm�1. InAPSD, four peaks on the ring of 8mm�1 indicate thatthe two symmetrical pairs of flutes pass on the sur-face and leave the aliased spatial frequency with8mm�1. As stated above, the spatial frequency of192mm�1 in the tool path is from the hard grain withthe frequency assumed as 32Hz.

The APSD bridges the gap between vibrationsand the surface topography generation; the analysisreveals that the vibration contradicting the surfaceerror is simply observed from the amplitudes. It isextremely efficacious to analyse the vibration sourceand shape details on the surface using the ASPD.

4.3 Significance of analysis on surface texture

Surface texture expresses the topography of a realsurface composed of certain deviations from theideal surface. Since the machined surface alwayscontains the anisotropic flutes, the profile methodsbecome invalid for the surface analysis. Com-paratively speaking, it is more efficacious andimpartial to employ the areal surface texture para-meters to investigate the whole surface data onthe evaluated surface area. For instance, the rootmean square height of the surface Sq is one of thecommon roughness parameters (ISO/DIS 25178-2)being used

Sq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1M ·N

PNj¼ 1

PMi¼ 1

z2ðxi; yjÞs

ð23Þ

To quantitatively analyse the effects of the variousfactors on the surface generation, the surface texturesignificance Sigqx is employed in the following for-mation

Sigqx ¼ jSqa � SqxjSqa

· 100% ð24Þ

Table 3 shows the significance of the main factorsin the dynamic cutting process on surface roughness,based on the data listed in Table 2. The valuesillustrate that dynamic cutting factors have a highinfluence on the surface generation in the ultra-precision cutting process.

1. The discontinuity of materials makes a largecontribution to the surface finish in nano/microcutting. The most significant factors are causedby hard grain and shear angle oscillation. The

Fig. 14 Areal power spectral density of the simulated surface

Table 3 Significance of dynamic factors on surfaceroughness

Factors Sigqx (%)

Workpiece materialproperty

Hard grain 29.30Shear angle oscillation 7.39

Cutting process BUE 12.67Regenerative vibration 6.63Tool wear 5.82

Machine toolperformance

Slideway side stiffness 5.17

Environmental vibration 9.86Spindle axial runout 3.74

Remainder 19.42



influences are clearly reflected in the PSD andAPSD amplitudes. Especially the hard grain leadsto the remarkable high-order harmonics in thecutting force and tool path, and thus composesthe main surface texture. Applying the singlecrystal material is the most effective approach toimprove the surface finish.

2. Cutting process dynamics is the second group tocontribute to the surface texture. Coolant willhelp to reduce the BUE and tool wear, and theregenerative vibration can be suppressed bycontrol of the previous residual height on thesurface texture.

3. The interesting finding is that the spindle axialrunout devotes even less than 4 per cent to thesurface roughness. Comparing this with the PSDfigure of the tool path and surface profile, it isconcluded that the cause of the above result isthe spindle runout being sharply attenuated bythe ‘sampling process’.

4. All the dynamics factors represent more than80 per cent of the contributions to the surfaceroughness in terms of the significance. Toimprove the surface roughness in the system, it isrecommended that the non-linear factors, such asadopting the homogeneous material and makingthe cutting system stiffer, especially for thedynamic response, should be tightly controlled.

5 EXPERIMENTAL VERIFICATION

In order to evaluate the effectiveness of the dynamiccutting process model and the surface predictionmethod, the surface simulated is verified throughexperimental work. Figure 15(a) shows the measure-ment result of an Al-6061 specimen which ismachined at the conditions of 1000 r/min spindlespeed, 0.01mm/rev feedrate, 0.01mm depth of cut,and 0.508mm tool nose radius, measured using a

Fig. 15 Illustrative comparison of the machined and the simulated surface generated under the samecutting conditions



Zygo NV 5000 system. The measured surface locatesat the 28–29mm radius area where the tool marks arenearly in straight lines.

The machined surface is also simulated underthe same cutting conditions as shown in Fig. 15(b).It is found that the form and features of the

simulated surface are consistent with those ofthe machined surface. Different from the cuttingat the axis centre, the hard grains turn to high fre-quency since the cutting speed at 28–29mm radiusis divided by the average grain size 30mm. Its vibra-tion on the surface profile is accordingly shifted to

Fig. 16 APSD analysis on the machined and simulated surface



2.27mm�1 after the sampling process by the spindlerotation.

To verify the modelling, simulation, and analysismethod further, the sampled point data of the surfacefrom Zygo measurements are processed by APSD, asshown in Fig. 16(a), and the corresponding APSD ofthe simulated surface, as shown in Fig. 16(b). The twoAPSD figures are identical and the relevant non-linear factor is highlighted in Fig. 16(b). The ampli-tudes of the APSD at certain frequencies also presentthe significances of the factors in the surface gen-eration process, which also dovetail the significanceanalysis result, as discussed.

6 CONCLUSIONS

1. The nano/micro cutting process is combinedwith the material property, cutting chatter, tool-ing geometry, actuation servo capability, themachine mechanical structure, environmentalvibrations, etc. The integrated simulation systemcan help to give a better understanding of theintrinsic relationship between the dynamic cut-ting process and the surface generation. Thenon-linear factors are included to emulate thereal cutting conditions, such as hard grains of thematerial, BUE, regenerative vibration, tool wear,and spindle runout. Integrating the drive andcontrol system with the cutting systemmodellingprovides the flexibility to improve the controlsystem performance and thus ultimately leads tothe good surface quality that is desired.

2. The surface simulation algorithm should takeaccount of the tool interference effect. The toolinterference commonly appears when the lowfeed rate or big rake radius cutting tool is adopt-ed. The tool interference may help to clear thehigher remaining tool marks and generate thelower surface roughness rather than the poorsurface finish.

3. Both the frequency ratio method and samplingtheorem can interpret the surface topographyformation. The spindle rotational frequency fssamples the tool–workpiece relative vibrations tothe aliasing frequencies as a lowpass filter. Thecomponents with frequencies higher than fs/2are attenuated sharply in one radial sectionprofile.

4. The PSD method is effective for the componentanalysis in one radial section, while APSD is morepowerful in both radial and circumferentialdirections. APSD provides vivid descriptions ofthe vibration frequencies, geometric distribu-tions, and their amplitude proportions. From theAPSD analysis, the original vibration in the toolpath can be reconstructed.

5. Non-linear factors have a major effect on thenano/micro cutting process. The significance ofnon-linear factors in total is more than 80 percent. The workpiece material heterogeneity iscentral to the surface roughness source. The fine-crystal material will positively lead to a bettersurface.

6. In the direct drive system, the force disturbancestransmit into the tool–workpiece displacementsthrough the servo motor and the slideway.Therefore, the Z axis servo performance and theslideway stiffness become the most critical partsin the dynamic cutting system. The higher thedisturbance rejection, the better the surfacequality becomes. The hydrostatic direct drivesystem can increase the physical stiffness anddamping, and the higher gain in the velocity loopwill enhance the damping, of the servo controlsystem.

Currently, the authors are undertaking an optimiza-tion design of the machine structure and cuttingconditions to improve the performance of the nanoface turning process; the related experimental resultswill be presented in other papers in the near future.More studies on the well-designed cutting trials andsimulations potentially applied to free-form surfacesin multiaxis nano/micro machining will be sum-marized in the authors’ next study to further verifythe approach developed.

ACKNOWLEDGEMENTS

The authors wish to acknowledge the assistance andsupport of the EU Sixth Framework IP MASMICROProject (Contract Number: NMP2-CT-2004-500095-2). Thanks are extended to partners in its RTD 5subgroup in particular.

REFERENCES

1 Simoneau, A., Ng, E., and Elbestawi, M. A. Surfacedefects during microcutting. Int. J. Mach. Tools Mf, 2006,46, 1378–1387.

2 Ni, H., Elmadagli, M., and Alpas, A. T. Mechanicalproperties and microstructures of 1100 aluminum sub-jected to dry machining. Mater. Sci. Engng, 2004, 385,267–278.

3 Cardi, A. A., Firpi, H. A., Bement, M. T., and Liang, S. Y.Workpiece dynamic analysis and prediction duringchatter of turning process. Mech. Systems and SignalProcessing, 2008, 22(6), 1481–1494.

4 Fang, N. and Dewhurst, P. Slip-line modeling of built-up edge formation in machining. Int. J. Mech. Sci., 2005,47, 1079–1098.



5 Durazo-Cardenas, I., Shore, P., Luo, X., Jacklin, T.,Impey, S. A., and Cox, A. 3D characterisation of tool

wear whilst diamond turning silicon. Wear, 2007, 262,

340–349.6 Jeong, G.-B., Kim, D. H., and Jang, D. Y. Real time

monitoring and diagnosis system development in turn-

ing through measuring a roundness error based on

three-point method. Int. J. Mach. Tools Mf, 2005, 45,

1494–1503.7 Cheung, C. F. and Lee, W. B. Characterisation of

nanosurface generation in single-point diamond turn-

ing. Int. J. Mach Tools Mf, 2001, 41, 851–875.8 Cheung, C. F. and Lee, W. B. A theoretical and experi-

mental investigation of surface roughness formation in

ultra-precision diamond turning. Int. J. Mach. Tools Mf,

2000, 40, 979–1002.9 Cheung, C. F. and Lee, W. B. Modelling and simulation

of surface topography in ultra-precision diamond turn-

ing. Proc. Instn Mech. Engrs, Part B: J. Engineering

Manufacture, 2000, 214(B6), 463–480.10 Kim, D.-S., Chang, I.-C., and Kim, S.-W. Microscopic

topographical analysis of tool vibration effects on dia-

mond turned optical surfaces. Precision Engng, 2002,

26, 168–174.11 Lee, W. B. and Cheung, C. F. A dynamic surface topo-

graphy model for the prediction of nano-surface

generation in ultra-precision machining. Int. J. Mech.

Sci., 2001, 43, 961–991.12 Lee, W. B., Cheung, C. F., Li, J. G., To, S., Du, J. J., and

Yin, Z. Q. Development of a virtual machining and

inspection system for ultra-precision diamond turning.

Proc. IMechE, Part B: J. Engineering Manufacture, 2007,

221(B), 1153–1174.13 Luo, X., Cheng, K., and Ward, R. The effects of

machining process variables and tooling characteriza-

tions on the surface generation. Int. J. Advd Mfg Tech-

nol., 2005, 25(11–12), 1089–1097.14 Luo, X. and Cheng, K. Nonlinear effects in precision

machining of engineering materials. In Proceedings of

the American Society of Precision Engineering Annual

Meeting, Portland, Oregon, USA, 2003, pp. 489–493.15 Takasu, S., Masuda, M., and Nishiguchi, T. Influence of

study vibration with small amplitude upon surface

roughness in diamond machining. Ann. CIRP, 1985, 34,

463–467.16 Delta Tau Data Systems, Inc. Turbo PMAC User Man-

ual, 2006.17 Childs, T. H. C., Maekawa, K., Obikawa, T., and

Yamane, Y.Metal cutting: theory and applications, 2000,

Arnold, London.18 Schmidt, C., Frank, P., Weule, H., Schmidt, J., Yen,

Y. C., and Altan, T. Tool wear prediction and verifica-

tion in orthogonal cutting. In 6th CIRP International

Workshop on Modeling of machining operations,

Hamilton, Canada, 20, May 2003.19 Yusuf Altintas Manufacturing automation: metal

cutting mechanics, machine tool vibrations, and CNC

design, 2000 Cambridge University Press, Cambridge.20 Fan, X. and Miller, M. H. The application of an

empirical tool force model on vibration assisted cutting.

In Proceedings of the American Society of Precision

Engineerings Annual Meeting, St Louis, Missouri, USA,

2002, pp. 484–489.

APPENDIX

Notation

a, b integral and fractional part of the frequencyratio

A, B constants in the tool wear processAevx, Aevz, fev

amplitude and frequency of the environ-mental vibration

Asr, Asa amplitude of the spindle radial/axial runoutAu, fu amplitude and frequency of the shear angle

oscillationAV actual velocityAa, va amplitude and frequency of the rake angle

variation due to the generation and removalof BUE

At, AH incremental amplitude of the shear stressand hardness due to the hard grain

CV, CA command velocity and accelerationdc depth of cutDACout digital-to-analogue converter (DAC) outputeevx, eevz environmental vibrationsexz, ezx errors of the slidewaysesr, esa spindle radial and axial runoutE process activation energyf feed per revolutionfp new frequency spectrum emerging due to

the sampling theoremfr frequency ratiofs spindle rotational frequencyfv vibration frequency in the time domainfvs vibration frequency in the spatial domainFE following errorFx, Fz cutting force in the X/Z directionFr resultant cutting forceh undeformed chip thicknessHt hardness of the cutting tool materialIE integral of the position errorKff1, Kff2 cutting force coefficients at the flank faceKp, Ki, Kd

proportional, integral, and derivative gainsKsp, Ksv position-loop and velocity-loop scale fac-

torsKtc, Krc, Kfc

cutting force coefficients contributed by theshearing action in the cutting speed (tan-gential), normal (radial), and thrust (axial)directions respectively

Kte, Kre, Kfe

cutting force coefficients at the cutting edgein respective directions



Kxz, Kzx side-stiffness of X and Z slidewaysKvff, Kaff velocity and acceleration feedforward gainslw tool flank wear widthLt projected length of tool edge in the X

directionm cycles of the spindle rotation lost in the

samplingmin function to return the smallest elementn spindle speedNp total simulation pointsNr total number of the spindle rounds in

machiningNs total number of sectionsPul signal generator of pulse at regular intervalsR universal gas constantRw workpiece dimensionR0 tool nose radiusSq root mean square height of the surfaceSqa surface roughness Sq when all the factors

are active in the simulationSqx surface roughness Sq when only the factor x

is switched off in the simulationSaw signal generator of the sawtooth waveSigqx surface texture significance of x factors on

the surface roughness SqT spindle revolution periodTf cutting temperature in the tool flank zone

V, Vs cutting speed and sliding speedx, y, z remained tool profile points in the rectan-

gular coordinatexc, yc, zc tool nose points in the rectangular coordi-

natexsm, zsm servo motions of X and Z slidewaysxtp, ztp real tool path in the machine tool coordi-

nate system

b side clearance angleDct, Dcw variation of cutting thickness and cutting

widthDf tool profile spacingDu change of the shear angle from oscillationDa variation of the rake angle due to BUEDu simulated angular spacingDt, DH increment of the shear stress and hardness

due to the hard grainu, r, h remaining tool profile points in the polar

coordinateua, ra, ha tool nose points in the polar coordinateur intersection angle of two continuous tool

pathsut, rt, ht all tool profile points in the polar coordinatem friction angle coefficientv, fr, fa spindle angular speed and phase shift of

radial/axial runout



247 dynamic cutting process modelling and its impact on

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